How to move only one side of a cube in object mode - equivalent of stretching the cube in that direction without moving other faces
You can just move the face in Edit mode:

If you want to do it in Object mode, put the 3D cursor on the face that is supposed to be the pivot point:

go into the Transform Pivot Point panel and choose 3D Cursor (or keyboard shortcut . to get the pie menu:

then scale on the axis you want (SX here):